Suppose that \(\lambda \left|G_{x,B}\right|\)  is odd. Let \(\sigma \) 
be an involution of \(G\)  and let \(y\)  be any point such that \(y^{\sigma }\ne y \) . Then \(\sigma \)  permutes the \(\lambda \)  blocks incident with \(y,y^{\sigma } \) , and hence \(\sigma \)  fixes one of them, say \(C\) , as \(\lambda \)  is odd. Thus \(k\)  is even, since \(\left|G_{C} \right|=k\left|G_{y,C}\right|\)  and \(\left|G_{y,C}\right|\)  is odd,
and hence \(\lambda (k+1)\left|G_{x,B}\right|\)  is odd. Therefore, \(\left[G_{x}:X_{x}\right] \)  is a divisor of \(\left|\mathrm {Out(}X\mathrm {)}\right|_{2^{\prime }}\) , since \(\left|X_{x}\right|=\frac{\left|G_{x}\right|}{\left[ G_{x}:X_{x}\right] }=\frac{\lambda (k+1)\left|G_{x,B}\right|}{\left[ G_{x}:X_{x}\right] }\) . Since \(\left|\mathrm {Out(}X\mathrm {)}\right|_{2^{\prime }}\)  is less than, or equal to, \(nf\) , \(nf/2 \) , \(3f\)  or \(f\)  for \(X\cong PSL_{n}(q)\) , \(PSU_{n}(q^{1/2})\) , \(P\Omega _{8}^{+}(q)\)  or \(X\)  as in the remaining cases, respectively, by [1]},
Table 5.1.A, and as \(n\geqslant 3\)  by Remark REF , it follows that \(\left[ G_{x}:H_{x}\right] <1+ef\)  for any \(X\) .
Suppose that \(\lambda \left|G_{x,B}\right|\equiv 2 \pmod {4}\) , \(\left( \Phi _{ef}^{\ast }(p),\lambda \left|G_{x,B}\right|\right)>1 \) , and also that either \(X\cong PSU_{n}(q^{1/2})\) , with \(n\)  even, or \(X\cong P\Omega _{8}^{+}(q)\) . Assume that \(k\)  is odd. By [2]}, one of the following holds:
(i).
\(G_{x}\)  is a parabolic subgroup of \(G\) , and \(q\)  is even.
(ii).
\(G_{x}\)  is the stabilizer of a non-singular subspace of \(PG_{n-1}(q)\) , and \(q\)  is odd.
(iii).
\(G_{x}\)  is a \(\mathcal {C}_{4}\) -subgroup of \(G\) .
(iv).
\(G_{x}\)  stabilizes an orthogonal decomposition into \(t\)  subspaces each of dimension \(n/t\)  in its natural projective module \(V\) .
(v).
\(X\cong P\Omega _{8}^{+}(q)\) , \(q\)  is a prime such that \(q\equiv \pm 3 \pmod {8}\) , and \(X_{x}\)  is isomorphic either to \(\Omega _{8}^{+}(2)\) 
or to \(2^{3}\cdot 2^{6}\cdot PSL_{3}(2)\) .

Case (i) is ruled out by Proposition REF . Since \(\lambda \left|G_{x,B}\right|\) 
divides \(\left|G_{x}\right|\) , and since \(\left( \Phi _{ef}^{\ast }(p),\lambda \left|G_{x,B}\right|\right) >1\) , it follows that \(\left|G_{x}\right|\)  is divisible by a primitive prime divisor \(u\) 
of \(p^{ef}-1\) . Thus, case (iii) is excluded by [1]}, Section 4.4 and Proposition 5.2.15.(i).
Assume that case (ii) holds. If \(X\cong PSU_{n}(q^{1/2})\) , with \(n\)  even, then \(G_{x}\)  is the
stabilizer of a non-singular point of \(PG_{n-1}(q)\)  by [1]},
Proposition 4.1.4.(II), since \(u\mid \left|G_{x}\right|\) . Thus,
\(k^{2}=q^{\left( n-1\right) /2}\frac{q^{n/2}-1}{q^{1/2}+1}\) 
and hence \(f\equiv 0 \pmod {4}\) , as \(q^{\left( n-1\right) /2}\)  must be a
square and \(n\)  is even. So \(\frac{q^{n/2}-1}{q^{1/2}+1}\)  is even, as \(q\)  is odd, whereas \(k\) 
is odd.
If \(X\cong P\Omega _{8}^{+}(q)\) , then \(G_{x}\)  is either the stabilizer in \(G\)  of a
non-singular point, or the stabilizer in \(G\)  of a non-singular line of type \(-\)  of \(PG_{7}(q)\)  by [1]}, Propositions 4.1.6.(II), since \(q\)  is odd and \(\left( \Phi _{6f}^{\ast }(p),\lambda \left|G_{x,B}\right|\right) >1\) . In the former case, \(k^{2}=q^{3}(q^{2}-1)\frac{q^{2}+1}{2}\)  and hence \(q^{2}-1\)  is a square. So, \(q^{2}=z^{2}+1\)  for some positive integer \(z\) , but this contradicts [6]}, A3.1. Therefore, \(G_{x}\)  is the stabilizer in \(G\)  of a non-singular line of type \(-\)  of \(PG_{7}(q)\) , then
\(k^{2}=q^{6}\frac{(q-1)^{2}}{(4,q-1)}\left( q^{2}+1\right) \left(q^{2}+q+1\right).\) 
So, \(q^{2}+q+1\)  is a square, but this contradicts [6]}, A7.1.
Assume that case (iv) holds. Then \(u\mid \frac{n}{t}\)  by [1]}, Section 4.2, and hence \(1+ef\leqslant u\leqslant n/2\)  by [1]}, Proposition 5.2.15.(i). This is impossible, since \(n\geqslant 4\)  in both cases by Remark REF .
Finally, assume that case (v) occurs. Then \(\left|G_{x}\right|\leqslant 24\left|X_{x}\right|\)  and hence \(\left|P\Omega _{8}^{+}(q)\right|\leqslant 24^{3}\left|X_{x}\right|^{3}\) ,
as \(G_{x}\)  is a large subgroup of \(G\)  by Lemma REF (2). Then either \(X_{x}\cong \Omega _{8}^{+}(2)\)  and \(p=3,5\)  or 11, or \(X_{x}\cong 2^{3}\cdot 2^{6}\cdot PSL_{3}(2)\)  and \(p=3\) . However, all the cases are ruled out, as
they violate \(\left[ X:X_{x}\right] =k^{2}\) . Thus, \(k\)  is even and hence \(\lambda (k+1)\left|G_{x,B}\right|\equiv 2 \pmod {4}\) . Arguing as
above, we see that \(\left[ G_{x}:X_{x}\right] \mid \lambda (k+1)\left|G_{x,B}\right|\) , where \(\left[ G_{x}:X_{x}\right] =\frac{\left|\mathrm {Out(}X\mathrm {)}\right|}{c}\)  for some \(c \geqslant 1\)  such that \(4\nmid \frac{\left|\mathrm {Out(}X\mathrm {)}\right|}{c}\) .
If \(X\cong PSU_{n}(q^{1/2})\) , with \(n\)  even, then \(\frac{\left|\mathrm {Out(}X\mathrm {)}\right|}{c}\leqslant nf/4<1+(n-1)f\) , as \(\left|\mathrm {Out(}X\mathrm {)}\right|=(q^{1/2}+1,n)f\)  and as \(f\)  is even. Then \(\left[G_{x}:H_{x}\right] =\left|\mathrm {Out(}X\mathrm {)}\right|/c<1+(n-1)f\)  and we obtain the assertion in this case.
If \(X\cong P\Omega _{8}^{+}(q)\) , then \(\frac{\left|\mathrm {Out(}X\mathrm {)}\right|}{c}\leqslant 6f\) , as \(4\nmid \frac{\left|\mathrm {Out(}X\mathrm {)}\right|}{c}\) , and hence \(\left[ G_{x}:H_{x}\right] \leqslant 6f<1+6f\) . This completes the proof.
